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Analysis reveals the conditions under which two analytic functions in an open disk share three small functions, suggesting equivalence.
Small functions were defined in complex analysis and next in ultrametric analysis. Order of growth and type of growth were also defined in complex analysis and have a similar definition in ultrametric analysis. Here we compare these two notions in the same way, on a complete ultrametric algebraically closed field of characteristic $0$ such as ₚ. Small functions with respect to an entire function f were studied in several articles. Inside an ''open'' disk, small functions also exist. After a general study, here we examine how two analytic functions inside an open disk can share three small functions, ignoring multiplicity and we give sufficient conditions proving that these two functions are equal.
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Alain Escassut (2025) studied this question.
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